fissiary polyhedra

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

fissiary polyhedra

Postby dodecahedron » Wed Oct 24, 2012 12:26 am

Does anyone know where to find a list.
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Re: fissiary polyhedra

Postby Keiji » Sun Feb 03, 2013 3:59 pm

What's a fissiary polyhedron?

A Google search only comes straight back to this page!
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Re: fissiary polyhedra

Postby Klitzing » Mon Feb 04, 2013 12:40 am

first of all: its not called "fissiary" but "fissary".

then: where to find about that topic?
Its a term being used by hedrondude. It even can be looked up in his glossary, where he states:
Fissary - A polytope like object that has peaks or lower dimensional elements that coincide completely. Fissary polychora either have compound vertex figures or edge figures that can split into two or more components.


if "peak" in turn is not known: here his according definition:
Peak - An n-3 dimensional element of an n-dimensional polytope. Example: vertices of polyhedra, edges of polychora, faces of polytera. If they are somewhat sharp, they will feel pointy, for example - if you were in the fourth dimension holding a tesseract, the edges (its peaks) will feel pointy (like a cube's corners).


thus: the application of that term onto polyhedra asks for such objects which have coincident vertices, without having coincident higher elements, which still are polyhedra - i.e. do not fall apart into being a compound.

finally: figures with (arbitrary) coincident elements often are implicitely or explicitely excluded from polytopes. Other authors OTOH do allow these. One of the best knowns here is Branko Grünbaum (esp. Are your polyhedra the same as my polyhedra?. This is why I usually call such figures simply "Grünbaumian".

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Re: fissiary polyhedra

Postby ubersketch » Fri Dec 15, 2017 1:21 am

There's only 2 I know of.
Small complex icosidodecahedron and great complex icosidodecahedron.
(sorry for necroposting)
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Re: fissiary polyhedra

Postby Mecejide » Mon Mar 25, 2019 3:57 pm

Well, those two aren't actually fissary because they're compounds, and in fact they aren't even polyhedra because they're exotic. So no, there aren't any fissary polyhedra.
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Re: fissiary polyhedra

Postby Klitzing » Mon Mar 25, 2019 9:21 pm

No rather there are in fact 2 such non-exotic, fissary complex icosidodecahedra: their face circuit around a vertex being either 3-5-3-5-3-5-3-5-3-5 (wound up twice), i.e. having completely coincident edges all over, resp. 5-3/2-5-3/2-5-3/2-5-3/2-5-3/2 (where 3/2 is a retrograde triangle), again having completely coincident edges.

And alike there are 2 such non-exotic, fissary great complex icosidodecahedra: their face circuit around a vertex being either 5/2-3-5/2-3-5/2-3-5/2-3-5/2-3 (wound up twice), i.e. having completely coincident edges all over, resp. 3-5/3-3-5/3-3-5/3-3-5/3-3-5/3 (where 5/3 is a retrograde pentagram), again having completely coincident edges.

Both each look exactly the same, and then in fact also to the exotic versions you mentioned - which occurs, when you'd identify the coincident edges each - or to the compounds you also mentioned. Thus a total of 4 different figures, which all look alike, but belong to different abstract things, both for the small and for the great case.

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Re: fissiary polyhedra

Postby Mecejide » Mon Mar 25, 2019 10:33 pm

I still don’t see how they aren’t exotic—they have coincident ridges, after all.
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Re: fissiary polyhedra

Postby Klitzing » Tue Mar 26, 2019 5:51 am

These first 2 each have completely coincident elements (edges), yes. But they don't have non-dyadic ridges: Each of those edges coincides to exactly 1 triangle and 1 pentagon only (for the small version) resp. to 1 triangle and 1 pentagram only (for the great version). Thus they aren't exotic.
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Re: fissiary polyhedra

Postby Mecejide » Tue Mar 26, 2019 4:17 pm

Yes. I thought that's what "exotic" means.
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Re: fissiary polyhedra

Postby Klitzing » Wed Mar 27, 2019 12:38 pm

No, by PolyhedronDude's Glossery you'd have
Exotic - A "polytope" is exotic if there are any ridges that contain more than two facets. They are not considered as true polytopes.
i.e. when it is non-dyadic.

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